Question
the height off the ground, in feet, of a squirrel leaping from a tree branch is given by the function H(x)=-16x^2+24x+15, where x is the number of seconds after the squirrel leaps. How many seconds after leaping does the squirrel reach its maximum height?
Answers
Answered by
GPT-4o mini
To find the time at which the squirrel reaches its maximum height using the function \( H(x) = -16x^2 + 24x + 15 \), we need to identify the vertex of the parabola represented by this quadratic equation. The vertex form of a parabola can be determined using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \) in the quadratic equation.
In this case, the coefficients are:
- \( a = -16 \)
- \( b = 24 \)
Substituting these values into the formula gives:
\[
x = -\frac{24}{2 \times -16} = -\frac{24}{-32} = \frac{24}{32} = \frac{3}{4}
\]
Thus, the squirrel reaches its maximum height \( \frac{3}{4} \) seconds after leaping. Therefore, the final answer is:
\[
\frac{3}{4} \text{ seconds}
\]
\[
x = -\frac{b}{2a}
\]
where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \) in the quadratic equation.
In this case, the coefficients are:
- \( a = -16 \)
- \( b = 24 \)
Substituting these values into the formula gives:
\[
x = -\frac{24}{2 \times -16} = -\frac{24}{-32} = \frac{24}{32} = \frac{3}{4}
\]
Thus, the squirrel reaches its maximum height \( \frac{3}{4} \) seconds after leaping. Therefore, the final answer is:
\[
\frac{3}{4} \text{ seconds}
\]